Clearly, there are all three elements in Orwell's Animal Farm. With such characters such as Napoleon, who represents Stalin, and Snowball, who is representative of Lenin, Orwell's narrative in an allegory of the communist Soviet Union. In this allegory, Orwell examines and satirizes the subversion and manipulation of the meaning of words that Communist Russia promulgated. Squealer, who "could turn black into white," is the propagandist.

The "fairy tale" is really a masque for the biting political satire of the Communist regime.

A fable has animal characters who have the physical characteristics of animals but the ability to think, feel, and speak like humans. To that extent, this is a fable. It's hard to claim Orwell wrote this as a disconnected story without any relevance to actual historical events. The fact that no one wanted to risk publishing it at first speaks of the power of this short little fable.

This is very hotly debated. I don't really see the point in arguing that Orwell did not intend for the book to be an allegory. By subtitling it "a fairy story" he sends a message to the readers that the book is allegorical. Although there are specific historical people and events, that does not mean the book is not also allegorical. The events, and sequence of events, my be based on fact but the message can be applied to any utopias or dictatorships.

There are certainly aspects of all three in this story. Since you ask specifically about fables, I'll talk about that.

A fable tends to involve animals acting in anthropomorphized ways. The things the animals do teach us something about our own lives, societies, etc. In this book, the animals' actions teach us a number of things. We learn, most importantly, that it is very difficult (perhaps impossible) to create a utopia in which no one will be exploited. This is one of the morals of this story.

So, since you have animals acting like people and a story that teaches us a lesson, you have aspects of a fable in this book.